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How to factor this conditional probability?

Cross Validated Asked by user292136 on December 8, 2021

In chapter 5.4 in the book Bayesian Data Analysis by Gelman et al. I see the following
expression related to a hierarchical model: $$p(mu,tau|y) propto p(mu,tau)p(y|mu,tau)$$
How do I derive this expression?

One Answer

Bayes rule says that: $$p(mu,tau|y)=frac{p(y|mu,tau)p(mu,tau)}{p(y)}$$

which means $p(mu,tau|y)propto p(y|mu,tau)p(mu,tau)$, i.e. LHS is proportional to the RHS.

This is typically done in Bayesian analyses where you're interested in the form of the posterior distribution and get rid of the constant part that doesn't change wrt the RVs of interest. For example, a subsequent analysis might be interested in calculating MAP estimates for $mu,tau$ by maximizing the posterior, in which case $p(y)$ has no effect.

Answered by gunes on December 8, 2021

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